A373285 Numbers k that are composite and not a powers of a prime k such that sopf^{h+1}(k) divides sopf^{h}(k), with sopf^{0}(k)=k, for h=0..A321944(k)-1, where sopf^{h} is the h-th iteration of sopf and sopf = A008472.

528, 1056, 1275, 1584, 2112, 2275, 2565, 3168, 3213, 3825, 3850, 3861, 4224, 4590, 4752, 5152, 5808, 6336, 6375, 6688, 7072, 7695, 7700, 8448, 9065, 9180, 9504, 9639, 10304, 10878, 11328, 11375, 11475, 11583, 11616, 12672, 12825, 13376, 13770, 14144, 14256, 15400, 15925, 16709, 16896

Offset: 1

Author

Rafik Khalfi, May 30 2024

Keywords

• nonn

Examples

			For k = 11475 = 3^3 * 5^2 * 17, sopf(k)=25 divides k and sopf(sopf(k))=5 divides sopf(k).
		

Crossrefs

Cf. A008472 (sopf), A321944.

Programs

Maple

f := proc (n)
    add(d, d = numtheory[factorset](n))
end proc:
h := proc (n)
    option remember;
    if isprime(n) then
        1
    else
        1+h(convert(numtheory[factorset](n), `+`))    end if:
end proc:
checkDivisibility := proc (n)
    local k, fk, fk1, result:
    result := true:
    fk := n;
    for k from 0 to h(n)-1 do
        fk1 := f(fk);
        if fk1 = 0 or `mod`(fk, fk1) <> 0 then
            result := false:
            break:
        end if:
        fk := fk1:
    end do:
    return result:
end proc:
g := proc (n)
    nops(numtheory[factorset](n)):
end proc:
findNumbers := proc (upper_limit)
    local n, results:
    results := []:
    for n from 2 to upper_limit do
        if checkDivisibility(n) and 2 <= g(n) then
            results := [op(results), n]:
        end if:
    end do:
    return results:
end proc:
upper_limit := 10000:
numbers := findNumbers(upper_limit);

Mathematica

s[n_] := DivisorSum[n, # &, PrimeQ[#] &]; q[n_] := !PrimePowerQ[n] && AllTrue[Ratios@ Reverse@ FixedPointList[s, n], IntegerQ]; Select[Range[2, 17000], q] (* Amiram Eldar, May 30 2024 *)